1993 Marino TAC IM Feedback Linearization

8/6/2019 1993 Marino TAC IM Feedback Linearization

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IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 38, NO. 2, FEBRUARY 1993

Adaptive Inp ut-Output LinearizingControl of Induction MotorsRiccardo Marino, Sergei Peresada, and Paolo Valigi

Abstract-A nonlinear adaptive state feedback input-outputlinearizing control is designed for a fifth-order model of aninduction motor which includes both electrical and m echanicaldynamics under the assumptions of linear magnetic circuits.The control algorithm contains a nonlinear identification schemewhich asymptotically tracks the true values of the load torquean d rotor resistance which are assumed to be constant butunknown. Once those p arameters are identified, the two controlgoals of regulating rotor speed and rotor flux amplitudeare decoupled, so that power efficiency can be improved with-out affecting speed regulation. Full state measurements arerequired.

I. INT RODUCT IONN the last decade, significant advances have been mad eI n the theory of nonlinear state feedback control (see1151 and [39] for a comprehensive introduction to nonlin-ear geometric control): in particular feedback lineariza-tion and input-ou tput decoup ling techniqu es have proveduseful in applications and applied even before the theorywas fully developed.The technique of state feed back linearization [17], [13]was developed in the effort of designing an autopilot forhelicopters [37]. It requires m easurem ents of the s tatevector x and knowledge of the parameter vector p inorder to transform a multiinput nonlinear control system

(x E R " , U E R " , p E R q )

into a linear and controllable one ( z E R " , v E R " )i =A z + Bv (2 )

Manuscript received June 1, 1990; revised May 17, 1991 and April 6,1992.Paper recommended by G. C. Verghese. This work was supportedin part by Ministero della Universit; e della Ricerca Scientifica eTecnologica.R. Marino and P. Valigi are with the Dipartimento di IngegneriaElettronica, Seconda Universit; di Roma, Via 0. Raimondo, 00173Rome, Italy.S . Peresada is with the Department of Electrical Engineering, KievPolytechnical Institute, Prospect Pobedy, 37 Kiev 252056 USSR.IEEE Log Number 9205180.

by means of nonlinear state feedback

(with B ( x , p ) a nonsingular m X m matrix V p E R9 ) an dnonlinear state space change of coordinates

Linear control techniques can then be applied in thedesign of the control U in (2). Necessary and sufficientconditions were determined in [17] and [13] for a system(1 ) to be locally feedback linearizable, i.e., transformableinto (2) via (3) and (4) n a neighborhood of x,.They arerather restrictive from a mathe matical po int of view. Nev-ertheless, they apply to detailed models of helicopters[371, synchronous generators [311, switched reluctancemotors [14] and permanent magnet stepper motors [SI.Electromechanical systems are good candidates for non-linear state feedback design since nonlinearities are oftensignificant and exactly known being modeled on the basisof physical principles.Whenever o utputs to be controlled a re defined as

y = h(x), y E R" (5 )the problem of making the input-output map decoupledand linear by state feedback (3) has been addressed andsolved in [111 and [16], following earlier applications inrobotics. The decoupling state feedback may ren der som estates unobservable from the outputs.Applications clearly indicate that even though nonlin-earities may be exactly modeled, the physical parametersinvolved are most often not precisely known. This moti-vated further studies on adaptive versions of feedbacklinearization and input-o utput linearization, since cancel-lations of nonlinearities containing parameters arerequired. For systems (1) which are linear with respect tothe unknown parameters p , sufficient conditions foradaptive feedback linearization were developed in [38]under sector type restrictions on certain nonlinearitiesand in [441, [181, and [201 under structural conditionswhich do not restrict the type of nonlinearities. Differentapproaches for nonlinear adaptive stabilization can befound in the survey paper [41]. Adaptive input-outputlinearization was studied in [43] und er global Lipschitzconditions on the nonlinearities multiplying unknownparameters. More recently, the more difficult problem of

0018-9286/93$03.00 0 1993 IEEE

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MARINO et al . : ADAPTIVE INPUT-OUTPUT L I N E A R I Z I N G CONTROL OF INDUCTION MOTORS 209

output feedback adaptive control has been addressed in[35] and [19] for single outp ut nonlinear systems.Even before the theory of nonlinear feedback controlwas fully developed, nonlinear state space change of coor-dinates (4) and nonlinear state feedback (3) were pro-posed in [31 and [41 for induction mo tor control in ord er toachieve an asymptotic decoupling in the control of speedand flux amplitude (the so called field oriented control).Nested loops of stand ard PI regulators are used in [27] toachieve robustness versus parameter variations. In orderto counteract uncertainties linear optimal control tech-niques and linear model reference adaptive techniqueshave also been proposed in [ll and in [281, [61, respec-tively. Decoupling is obtained only in steady state, i.e.,when the flux amplitude is kept constant. Coupling is stillpresent when flux is weakened in order to operate themotor a t higher speed within the input voltage saturationlimits [27, p. 2171 or when flux is adjusted in order tomaxim ize pow er e fficiency [26], [22]. A different approachwhich makes use of variable structure techniques and ofnew sta te coor dinat es was proposed in [42]. Nonadaptiveinput-output decoupling controls were present ed in [29],[30], and [25] using geo me tric techniques (see also [21]). In[25], a fifth-order model which includes the mechanicalpart is used: exact decoupling in the control of speed andflux amplitude is achieved by a static state feedbackcontrolle r. I n [291 an d [301, a simplified model is used:only the electromagnetic part is modeled assuming thespeed of a slowly varying parame ter. E xact decoupling inthe control of electric torq ue a nd flux amplitude using theamplitude and the frequency of the voltage supply asinputs is obta ined in [29] by a dynamic (second o rder )compensator and in [30] by a static state feedback com-pensator. An indirect adaptive version of the decouplingalgorithm p ropose d in [30] can b e found in [45], where twoelectrical pa rameters are assumed to be unknown.The main result of this paper is to develop an adaptiveversion of the con troller prese nted in [25], assuming thatload torque and rotor resistance are unknown but con-stant parameters. In Section I1 a fifth-order state spacemodel of an induction motor, which includes both electri-cal and mechanical dynamics, is given. In Section 111previous control schemes are reviewed and it is shownthat field oriented control can be viewed as a feedbacktransformation which achieves asymptotic input-outputdecoupling and linearization. It is also established that themodel is not state feedback linearizable and that thedynamics made unobservable by a state feedback input-output linearizing control (zero dynamics) are due to therotation of the flux vector. In Section IV an adaptiveversion of the exact decoupling and linearizing controlgiven in [251 is developed, which covers the mo re realisticsituation in which the load t orque a nd the rotor resistanceare not known. Rotor resistance may have a range ofvariation of +5 0% around its nominal value due to rotorheating. Even though available nonlinear adaptive resultsdo not apply as such to the model given in Section 11, thekey idea in 1181 leads to a second-order nonlinear identi-

fication schem e which asymptotically tracks the true valueof load torque and, when electric torque is different thanzero, the true value of rotor resistance as well. Theadaptive state-feedback linearizing control achieves fulldecoupling in speed and rotor flux magnitude regulationas soon as the identification scheme has converged to thetrue parameter values: this allows us to improve powerefficiency by adjusting flux levels, without affecting speedregulation. In Section V the effects of parameter uncer-tainties on the performance of nonadaptive decouplingcontrol are analyzed in detail and shown by simulations.Additional simulations illustrate the performance of theproposed adaptive control algorithm in speed and fluxamplitude regulation, showing that voltage supply wave-forms are implementable by actual inverters and currentsare within acceptable limits. The main drawback is th erequirement of flux measurements: however, preliminarysimulations repor ted in [36] indicate that the proposedadaptive control maintains good performances even whenrotor flux values a re provided by the observer given in [46]and driven by rotor resistance estimates provided by theadaptation law. Preliminary versions of this work wererepor ted in [33] and [34]. Additional simulation studiescan be found in [36].

11. IND UC TIO NOTORMODELThe reader is referred to [lo] and 1231 for the generaltheory of electric machines and induction motors, to [27]for related control problems, and to [91 for digital imple-mentations. The symbols used and their meaning arelisted in the Appendix.An induction motor is made by three stator windingsand three rotor windings. Krause and Thomas [241 intro-duced a two phase equivalent machine representation (seethe Appendix for the exact transformation of three phasevariables into two phase ones used in this paper) with tworotor windings and two stator windings. Their dynamics

are described byd*SOR,i,, +- U,,dtd*SbR,i,, +- uS bdt

(7 )

where R , , a) , U, denote resistance, current, flux linkage,and stator voltage input to the machine; the subscripts sand r stand for stator and rotor, ( a ,b) denote the compo-nents of a.ve ctor with respect to a fixed stato r referenceframe, ( d ' , q ' ) denote the components of a vector withrespect to a fram e rotating at spee d n p w ;and np denotesthe number of pole pairs of the induction machine and o


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210 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 38, NO. 2, FEBRUARY 1993

the rotor speed. Let 6 denote an angle such thatd 6_ -t - n P w , 6(0) = 0.

We now transform the vectors ( i r d , , , ,), @ r d t , @r q . ) in therotating frame ( d ' , q ' > into vectors ?irU,r b ) , ( $ r , , r,,) inthe stationary frame (a,b ) by

Applying transform ations (91, (10) and using equation (8),(6) and (7) becomed*s,

d*SbR, i , , +- U, ,dt

dt, i , , +- U , hd*ruR,i,, +- n pw&,, = 0dt

Under the assumptions of linearity of the magnetic cir-cuits and of equal mutual inductances and neglecting ironlosses, the magnetic equations are (see [23, p. 1721)Ccsa = ' s i , , + Mi,,@sh = L s i s b +$ru = M'su + r i r u$r b = + L r i r b (12)

where L,, L , are autoinductances and M is the mutualinductance; as we shall see, the assumption of linearitywill be enforced by a control action which will keep theabsolute value of the rotor flux below the nominal value.The purpose of introducing the transformations (9) an d(10) is precisely to obtain (12) which is independent of 6:in fact fluxes and currents in (6) and (7) are related by6-dependent auto and mutual inductances.Eliminating i,,, i,, an d qbSu71+9,,, in (11) by using (12),we obtain

> U

R,i, , + --lC' ,b + ( L , - g)% = U , , ,L , dt

The torque produced by the machine is expressed in

terms of rotor fluxes and stator currents as

L r

so that the rotor dynamics are

where J is the moment of inertia of the rotor and of anytool attached to it and TL is the load torque.By adding the rotor dynamics (15) to the electromag-netic dynamics (13) and rearranging the equations in statespace form, the overall dynamics of an induction motorunder the assumptions of equal mutual inductances andlinear magnetic circuit a re given by the fifth-order model:d w n,,M T Ldt JL, J- ( * r a i s b - $r bi s o) - -

where i , +,uJ denote current, flux linkage and statorvoltage input to the machine; the subscripts s and r standfor stator and rotor; ( a , b ) denote the components of avector with respect to a fixed stator reference frame and

From now on we will drop the subscripts r an d s sincewe will only use rotor fluxes (+,,, q5rb) and stator currents(i,,, is,) as state variables. Let

= 1 - ( M ~ / L , L , ) .

= ( $0, @b 7 i b ) T (17)

(18)be the state vector and let

P = ( ~ 1 3 ~ 2 ) ~( T L - T L N ~ R ~',,ITbe the unknown parameter deviations from the nomi-nal values TLN an d R ,, of load torque TL and rotorresistance R , . TL is typically unknown whereas R , mayhave a range of variations of *50% around its nomi-nal value (see [27, p. 2241) due to rotor heating. Let U =(uU,uh)' be the control vector. Let (Y = ( R r , , , / L r ) ,p =( M / w L , L , ) , y = ( M 2 R , , / ~ L , L : ) + ( R , / ( T L , ) ,p = ( n ,M / J L r ) , be a reparameterization of the inductionmotor model, where a , p, y , p are known parameters



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MARINO et al.: ADAPTIVE INPUT-OUTPUT LINEARIZING CONTROL O F INDUCTION MOTORS 211

1 '

0 '

_ _J000

depending on the nom inal value R I N . ystem (16) can berewritten in compact form as Since cos p = ($,/I I) I), sin p = ( & / I $I>, with I $ / =d m , ro m (24) an d (25) we have= f ( x ) uaga + ubgb 'Plfl +P2f2( ' ) (19) $ai, + $bibl d = I * Ihere the vector fields f , g,, gb, f l , 2 are

(24)

(25)

cos p sin p(j: j = [ -sin p cos p ] (ja)'

f d x ) =

i2- n p w i , - ( Y M ~a p q d + v d+ npw id + a M - + U, (30)

*d i d i ,np

f 2 ( x ) =

We now reinterpret f ield oriented control as a statefeedback transformation (involving state space change ofcoordinates and nonlinear state feedback) into a controlsystem of simpler structure. Defining the state space(21) change of coordinateso = w

$ d = &&-??*b*a

p = arctan -$sib - $bia1, = I * I

and the state feedback-1

the system (16) becomesd w T L_ -- p$diq -7dt

111. INDUCTIONMOTORCONTROLA. Field Oriented Control - - + a M i dd- -dt

A classical control technique for induction motors is byBlaschke 131, [4] in 1971, it involves the transformation ofthe vectors ( i a , b ) , +a , in the fixed stator frame ( a ,b)into vectors in a frame ( d ,q )which rotate along with theflux vector (+,, defining

did- - y i d + f f p & + n p w i , + a M 4 + -udi,i, 1_ - - i , - pnpwt+!td- n p w i d- a M - + -U,i,dt

(29)

i2now the field oriented control. First introduced by dt *d u L s

*d uL r-P = n p w + ~ M L .(23) dt *d*b

*ap = arctan -

Defining the nonlinear state feedback controlthe transformations are


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212 IEEE TRANSACTIONS ON AUTOMATIC CONTROL,VOL. 38, NO. 2, FEBRUARY 1993

so that ( 28) becomes

the following closed-loop systemd o TLNdtd i ,dt

_ -- p@di9 -_ -- - i , + U,

If w an d thd are defined as outputs, field oriented controlachieves asymptotic input-ou tput linearization anddecoupling via the nonlinear state feedback (281, (341,(36) :PI controllers are then used to counteract parametervariations.During flux transient the nonlinear term i,bdiq in ( 32)makes the first four equations in ( 32) still nonlinear andcoupled: as a result speed transients a re difficult to evalu-ate a nd may be unsatisfactory, Flux transients occur whenthe m otor has to be operated above the nominal speed: inthis case flux weakening (for instance = ( k / w r e f ) )srequired in order to keep applied voltage within inverterceiling limits [27, p. 2171. Even when the motor is oper-ated below the nominal spee d, flux may be varied in orde rto maximize power efficiency (see [26] , 22]) .B. Input-Output DecouplingAs shown in [25] see also [21]) , ield oriented controlcan be improved by achieving exact input-output decou-pling and linearization via a nonlinear state feedbackcontrol which is not more complex than (31). We nowsummarize this technique which will be made adaptive in

next section. The following notation is used for the direc-tional (or Lie) derivative of state function +(XI: R" + Ralong a vector field f ( x ) = ( f , ( x ) ; - . , n ( x ) )

- - f f $ b d + a M i dd- -dtd id- - y i d + U,dtdP i ( 3 2 )n p w + ~ M X

is obtained. System ( 16) is transformed into ( 32) by thefeedback transformation (271, (31 ).dynamics are linear

dt *d

( 3 7 )J+~ f $ C G f i ( x ) .System ( 32) has a simpler structure: flux amplitude i = 1Iteratively, we define L$$ = L f ( L ' f -)$I.Define the change of coordinates (see also [ 4 2 ] )The outputs to be controlled are w an d t + i,!~,'.- -d*d - -a+d + a M i ddt

dtdid Y , = M X ) = w

T L N( 3 3 )-- -y id + U,Y 2 = L f + l ( x ) = p.(cGhib- * b i a ) --nd can be independ ently controlled by for instance via JY3 = $ A x ) = * + *PI controller, as proposed in [27]

= - k d l ( * d - * d r r e f ) - d Z / f ( * d ( T ) - q d r e f ) d T .0( 3 4 )

When the flux amplitude $d is regulated to the constantreference value I , ! J ~ ~ ~ ~ ,otor speed dynamics are also lin-eard w TL_ -- p*dref i q - -t Jdi,dt_ -- - i , + U, ( 3 5 )

y , = arctan (t= $3which is one to one in Q = (x E R5: 2 + @ # O} but itis onto only for y 3 > 0, -90 s y , I 90. The inversetransformation isw = Y l+a = & OS Y ,* h =6 in Y ,

1 Y + 2 a Y , 1 TLNnd can be independently controlled by U, , for instance bytwo nested loops of PI controllers, as proposed in [271 i , = -(COSY5( 4 2 a M ) - - p Y 5 ( Y 2 +7))6



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The dynamics of the induction motor with nominal para-meters are given in new coordinates byY l = Y 2

Y 3 = Y 4

Y 5 = L f 4 3 . (40)

9 2 = L2f41 + LgaLf l U a + LgbLf l U b

Y 4 = L2f42+ LgaLf4 2 u a + LgbLf 4 2 u b

The first four equations in (40) can be rewritten as

- P n p o ( r C d i a + * b i b )L2fc#J2 ( 4 2 + 2 a 2 p M ) ( *2 + *+ 2 a M n p o ( Gaib- & , i n )

- ( 6 a 2 M + 2a')'M)( +aia + + b i b )+ 2a2M2( i , 2 + i,")and D(x ) s the decoupling matrix defined as[ g k f ' 1 LgbLf " ' 1( x ) =

L g k f 42 Lg&f 42P P-- *b Z * aULs

2 a M= [ ? @a -$ bc+LS

Since

D ( x ) s nonsingular everywhere in Q .The dynamics of the flux angle y , = 4 , ( x> area M

= n p o + =( *sib - * b i a )d4 3 d Y 5- - - -dr dr(45)

The difference between flux angular speed C$3 and rotorspeed n pw is usually called slip speed, ws ,which can beexpressed, recalling the expression of a , as

R r N M +sib - $ b ia4, - n p w = os=-, *2 + *

The input-output linearizing feedback for system (40)is given by

where U = (ua, b)T is the new input vector. Substituting(47) in (40) the closed-loop dynamics become in y-coordi-nates

Y1 = Y 23 2 = uaY 3 = Y 4Y 4 = ub

Equations (45) or (46) represent the dynamics which havebeen made unobservable from the outputs o nd $2 + +by the state feedback control (47). In order to trackdesired smooth reference signals or, , (?)nd I +fe f ( t ) fo rthe speed y 1 = w and the square of the flux modulusy , = 42 + &, the input signals U, an d q, n (47) aredesigned asv a = - k a l ( Y l - or e f ( t ) ) - a 2 ( ~ 2- 4 e f ( t ) ) + &ref ('1

= - k a l ( - orer - a2 ( P( *sib - ( l b ia )= - k , i ( y 3 - I+IL)- b 2 ( y 4 - I4IL) + Ili;I:ef= -k b l ( lG b 2 + - +l:ef) - b 2 ( 2 a ( M ( $ a i a + * b i b )

-(*; + + - 4I:ef) + Ili;Ifef (49)where ( k a , , a 2 ) nd ( k b l , b 2 ) re constant design param-eters to be determined in order to make the decoupled,linear secon d-order systemsd 2 dz(- o r e f ) = - k a 1 ( - wref) - a 2 ~ ( - orer)

d 2- ( I + I ~ - +I:ef) = - k b 1 ( l + l 2 - qItef )dr 2d

- b 2 z ( l $ 1 2 - +I:ef) (50)asymptotically stable and to shape their responses.

Remarks:1) The closed-loop system (48) is input-outputdecoupled and linear: the input-output map consists of apair of second-order systems. This allows for an indepen-dent regulation (or tracking) of the outputs according to(50). Transient responses are now decoupled also whenis varied, even independently of o r e f .his is an


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214 IEEE TRANSACTIONS ON AUTOMATIC CONTROL. VOL. 38, NO. 2, FEBRUARY 1993

improvement over the field oriented control (see also[251).2) Stat e space change of coordinates both in the fieldoriented control and in the decoupling control [i.e., (27)and (39)] are valid in the open set R = { x E R': $2 +$ # 0); notice that + + + = 0 is a physical singularityof the motor in starting conditions.3) As in field oriented control, while measurementsof ( w , ,, i,) are available, measurements of ($,, $b )require installing flux sensing coils or Hall effect trans-ducers in the stator which is not realistic in general-purpose squirrel cage machines.4) Easy computations show that the induction motormodel (16) is not feedback linearizable. Th e necessary andsufficient conditions given in 1171 fail; in fact the distribu-tion gi = span {ga,g,, ad fg,, ad,g b} is not involutivesince the vector field [adfg,, adfg,] does not belong to 27,(ad ,Y or [X ,Y ] denotes the Lie bracket of two vectorfields; one defines recursively ad kY = ad,(ad;l ' Y ) ) . ol -lowing the results in [32], since F0= span {g , , gh) is invo-lutive and rank FI = 4, it turns out that the largest feed-back linearizable subsystem has dimension 4. This shows

thatthe control (47), (49) provides the largest linearizablesubsystem in the closed loop.5 ) The state feedback control (47), (49) is essentiallythe o ne proposed in [25]. Th e only additional contributionin this section is to make clear that t he decoupling controlmakes the angle +3 unobservable from the outputs andthat (16) is not feedback linearizable: this will be impor-tant in the design of an adaptive control in next section.Exact input-output decoupling controls for inductionmotors are also proposed in [29], [30] with reference to asimplified model: the mechanical dynamics in (16) are notconsidered and w is viewed as a parameter in the lastfour equations of (16). In [29] the inputs are constrainedto be of the formU, = v c o s 0ub = Vsin 0 (51)

an d two integrators are addedd V- U Idtd 0 -- U?

so that (ul,U , ) are the new inputs. The outputs are chosento be the electric torque ( n , M / L , ) (+,i, - and therotor flux amplitude square $2 + 0;. A nonlinear statefeedback control for uI and u2 is obtai ned in [291 whichachieves input-output decoupling and linearization whilemaking a two dimensional state space submanifold unob-servable from the outputs. In [30] flux and currents areexpressed in a rotating frame according to the transforma-tions (24) and (251, where the dynamics of p ( t ) are

controlled by a third input U, so thatd P i- n, ]w + ~ M LU,.dt *d (53)

The model consists of five states ( i d , i,, qd, )~,) and isdriven by three controls. A nonlinear state feedback con-trol is designed which decouples and linearizes the threecontrol actions of regulating torque, rotor flux amplitudeand of forcing the reference frame ( d , q ) to rotate alongwith the rotor flux vector while making p unobservablefrom the outputs. Assuming two electrical parametersunknown, an indirect adaptive version of this con trol wasdeveloped and simulated in 1451.An input-output decoupling control law in the d - qreference frame has also been proposed in [211. Thecontrol algorithm is based on flux estimates provided byan open-loop flux simulator. Following [12], rotor resis-tance errors are computed in [21] on the basis of steady-state regulation errors and used in the output feedbackcontrol algorithm in order to reduce the steady-stateregulation errors. The performances of the overall systemare verified by simulations and experimental tests.

IV . ADAPTIVEN P U T - O U T P U TINEAR IZATIONWhen a decoupling control algorithm (47), (49) is used,variations in load torque TL and rotor resistance R, causeloss of input -outpu t decouplin g, steady-sta te trackingerrors and deteriorated transient responses. This calls foran adaptive version of (47), (49) which will be developedin this section under the assumptions that TL and R, areunknown constant parameters. In this section an adaptivetracking problem is addressed for a class of referencesignals satisfying the following assumptions.Assumption I : The reference signals wJ t ) an d I$l;ef

( t ) are required to be C 2 bounded functions with boundedderivatives such thatlim wrCf t ) = c , ,t'X

with c , ,c2 E R .Let us rewrite system (19) in the y-coordinates definedby (38); since the Lie derivatives L f Z 4 1 , f , L f 4 1 ,L f , & ,zero, we haveLf1L,+2, f 1 4 3 >f 1 L f 2 4 2 ?g"439L g h 4 3 are all equal to

Y l = Y2 + P l L f l 4 I



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where

T

6 a L , M L , + 2 M 3( $ a i , + $bib>?- UL,L)

( 5 5 )

An adaptive version of input-output linearizing controlswas proposed in [43],which requires an overparameteriz-ation and global Lipschitz property for the nonlinearitiesmultiplying the parameters. Adaptive versions of feedbacklinearizing controls were developed in [381 under sectortype restrictions on certain nonlinearities, in [44]understructural matching conditions, in [18] under extendedmatching conditions and, more generally, in [20],underpure feedback conditions which do not require Lipschitzor sector type restrictions. No one of the above tech-niques apply in our case since the nonlinearities involvedare not globally Lipschitz and the system is not feedbacklinearizable . How ever, we will use the adapt ation te ch-nique proposed in [18] under the so called extendedmatching structural condition and show directly the con-vergence both of tracking errors and of parameterestimation errors.Let 8 0 )= (b,(t>,i2( t )>T e a time-varying estimate ofthe parameters and let

be the parameter error. Following [18] we introduce atime-varying state space change of coordinates depending

on the parameters estimate @ ( t )21 = Y l2 2 = Y 2 +81Lf1412 4 = Y 4 +8 2 L f 2 4 22 3 =Y3

2 5 = y 5 .In z-coordinates system (19) becomes

2 , = 2 2 + e p , L f , 4 1i 2 = L2f4l + P 2 L f 2 L f 4 1+ Z L f 1 4 18 ,

2 = L2f42+ P 2 L f 2 L f 4 2+ 7 L f 2 4 22

+ L g k f l U a + L g h L f 4 l U bi 3 = 2 4 + e p 2 L f 2 4 2

+82LfLf242 82P2L2f242+ U0(Lg0Lf42B2Lg.Lr242)+ U b ( L g b L f 4 2 + 8 2 L g b L f 2 4 2 )

is = Lf43 +P2Lf243Let ( 2 ) (a]

(57)

1g h L f 4 1LgbLf42+ 82Lg,Lf242

0 1 0 1K a = [ - k a l - k a 2 ] . Kb = [ - k b l - k b 2 ] ( 6 2 )are asymptotically stable; w,&), I $ f e r ( t ) are referencesignals which satisfy Assum ption 1. Since


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216 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 38, NO. 2, EBRUARY 1993

the decoupling matrix is singular not only when($2 + $ = 0 as in the no nadaptive case but also when$ , ( t ) = -R,,,,; this additional singularity has to be takeninto account in the design of the adaptive algorithm.Define the reference m odels

Th e model reference tracking error is defined ase ( 2 ] - I M 7 ' 2 - 2 M 7 ' 3 - 3 M 9 ' 4 - 4 M ) (65)

and its dynamics are given byd l = e2 + e,,Lf,+l4 = -kulel - kuze2 + e,2LfzL f+l4 e4 + f3,,2Lf2+2

4 = Lf+3 + P 2 L f 9 3e4 = - kb le3 - b2e4 + ' p ? ( L f 2L f +2 ' $ 2 L ; 2 + 2 )

(66)with e i ( 0 )= 0, 1II .While the dynamics of z s are

the dynamics of the vector e can be rearranged as

whereK = block diag (K, ,Kh ) (69)

W ( z ,i2)s called th e regressor matrix and is a function ofthe x-variables (and therefore of the z-variables).Le t P = block diag(P,, P b ) be the positive definitesymmetric solution to th e Lyapunov equation

K T P + P K = - Q (70)with Q = block diag, Q,, an d Qb positive defi-

nite symmetric matrices. Consider the quadratic functionv = eTPe + e iT e , (71)

where r is a positive definite symm etric matrix. T he timederivative of V is

If we now define

or, equivalently,

which defines the dynamics of the parameter estimate@ ( t ) , nd use (701, then (72) becomesdVdt_ - -eTQe. (75)

This guarantees that e ( t ) and e,,, and therefore j X t > ,a rebounded and that e ( t ) s an L2 signal. Under Assumption1 fo r ( w r e f ( t ) ,$I?,,f(t)), it follows from asymptotic stabil-ity of (64) and from (65) that the first four state variables(z1;.-, z 4 ) are bounded. We are guaranteed to avoid thesingularities z 3 = 0 an d b2= - R r N for the decoupl-ing matrix, and therefore for the control (59) as well,when the initial conditions (e(0) = 0, e,(O)) are in S ={ ( e , ,) E R h : eTPe + e i r e , I, v > 01, the largest setentirely contained in ((e, e P )E R 6 : epz< R,, e3 > - c 2 } ,with c 2 given in Assumption 1. Since W ( z , 2 ) s continu-ous, contains only bounded functions of z5 (sine andcosine), and ( z l , z 2 , z 3 , 4 , 2 ) are bounded, it follows th atW(z , is bounded and therefore e and $ are boundedas well. Now, since e is a bounded L2 signal with bounde dderivative t , by Barbalat lemma ([40], p. 211) it followsthat

lim (1 e( ) 1 = 0, (7 6 )t'X

i.e., zero tracking error is achieved, both with respect tothe reference model and to the reference signals. Further-mor e, since, according to (671, is s bounded for ( e , e,) ES and the parameters are assumed to be constant, itfollows that e = ( d / d t ) [Ke + W( ,$ ) e ] is bounded aswell. Hence, e being bounded, t? is uniformly continuousan d (76) implies, by Barbalat lemma again, that2 . plim IIi( t ) 1 = 0, (77)t + =

therefore it must be



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MARINO el al . : ADAPTIVE INPUT-OUTPUT LINEARI ZING CONTROL OF INDUCTION MOTORS 217

Equation (78) implies, from (55) and (681, that1

t , t - m Jlim Lfl+le,,(t) = lim - -e,,(t) = 0,M 2Iim Lf,Lf+lep,( t ) lim -t - m t - m

= oi.e.,lim ep , ( t ) = 0,t - m

and, since by virtue of Assumption 1 l i m t - m T ( t ) = T L ,when TL f 0,lim e p , ( t ) = 0.I

Remark 6: The assumption that the initial conditionsbelong to the set S is due to the existence of singularpoints for the determinant (63). It follows that the largestz 3 ( 0 )= $I2(O) and R r m i , re the largest are the allowableinitial condition errors (e(O), e,(O)). It is therefore moreconvenient to start the adaptive control algorithm whenthe squared flux amplitude l $ I 2 is far from zero. On th eother end it can be seen from the model (16) itself andfrom (43) that large values for R , make the control taskeasier.In conclusion the results obtained can be summarizedas follows.Theorem: Consider th e closed-loop system given by theinduction motor model (19) and the adaptive dynamicstate feedback control (591, (641, (74). Ifa> the unknown parameters p , and p2 are constants,b) the reference signals (wre f ( t ) ,$lfef(t)) satisfyAssumption 1,c) the initial conditions (40 ) = 0, e,(O)> E S =( ( e ,e,,) E R 6 : eTPe + e;Te < v, v > O) , the largest

set entirely contained in (Le,) E R 6 : ep2< R , , e 3> -cJ,then:

lim ( o ( t ) q e f ( t ) ) = 0,lim ( h ( t )- href( t ) ) 0,

lim ( I $ ( t ) l - $ l r e f ( t ) ) = 07

(82)(83)(84)

t

t - m

t + m

whereMoreover if TL # 0, then

where

V. SIMULATIONSThe proposed control algorithm has been simulated fora 15 KW motor, with rated torque 70 Nm and rated speed220 rad/s, whose data are listed in the Appendix.

The simulation test involves the following operatingsequences: the unloaded motor is required to reach therated speed and the rated value of 1.3 Wb for rotor fluxamplitude ]$I, with the initial estimate of rotor resistanceR , in error of +50%. At t = 2 s. a 40 Nm load torque,which is unknown to t he controller, is applied. This impliesa reinitialization at t = 2 s. and the theorem given inprevious section a pplies from 0 to 2 s. and from 2 s. on. Att = 5 s. the speed is required to reach 300 rad/s., wellabove the nominal value, and rotor flux amplitude refer-ence is weakened according to the rule I$lref(t) =( k / q e f ( t ) ) . he reference signals for flux amplitud e andspeed, reported in Fig. 1, consist of step functions,smoothed by means of second-order polynomials. A smalltime delay at the beginning of the speed reference trajec-tory is introduced in order to avoid overlapping betweenflux and speed transients.Both nonadap tive (471, (49) and adaptiv e (59), (641, (74)control laws have been simulated, with TLN= 0 an d

In the nonadaptive case, we observe from simulations(compare Figs. 2, 3, and 4, where dashed lines stand forreference trajectories, solid lines correspond to simulatedbehavior and dotted line represents the electric torque)that the parameter errors cause a steady-state error bothin speed and flux tracking; they also cause a couplingbetween speed and rotor flux which is noticeable bothduring speed transient and at load insertion ( t = 2 s.). Inorder to clarify the effects of unknown parameters, con-sider the closed-loop system, obtained applying feedbackcontrol (471, (491, for the simple case of a regulationproblem, to the motor (19) [recall (5411:

R,N = 0.15 R.

i s = Lf43where 2 = ( C l , e,, e,, Z4IT = ( y l .- qef,, , y 3 ;I$l;ef, y 4 I T , s the regulation error, with qef nd I$I refconstant, K has the structure given in (62), (691, whileW * p takes into account the effects of parameter uncer-

Authorized licensed use limited to: Seun k u Park. Downloaded on June 2 2009 at 02:58 from IEEE X lore. Restrictions a l .


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218 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 38, NO. 2, FEBRUARY 1993:[TI~p Flux Am litude Referenceb'c) 200

100

0 2 4 6 8 0 2 4 6 8

v)

0 0.8Time (sec)ime (sec)(a) (b)

Fig. 1.

,...~~......

.............h-2- 1001v)

0 2 4 6 8

Flux Amplitude1.5 I

00 2 4 6 8Time (sec) Time (sec)

(a) (b)Fig. 3.

e

0 2 4 6 8 0 2 4 6 8Time (sec)ime (sec)

(a) (b)Fig. 4.

tainties [see (18)l. Matrix W* entries are given in (55) ,from which it is easy to see that Lj,+1 is constant andL J 2 L f 4 1 s proportional to the electric torque T . T heentry L f 2 + 2 s proportional, via a nonzero constant, to thederivative of the squared flux amplitude and therefore,once flux steady state is achieved, LjZ+2= 0. The entryL J 2 L J & . an be rewritten as L J 2 L J &= c,(d141~/dt)c 2 T 2 ,with c1 and c2 nonzero constants. When electrictorque is zero and flux amplitude steady state is achieved,LJ2LJ$b2= 0. U p t o 2 s., there is no load torque so that

I 00 2 4 6 8 0 2 4 6 8

Time (sec) Time (sec)(a ) (b)o,2 True & Estimated Resistanceo True & Estimated Load I I:20 I 0

( C ) (d)0 2 4 6 8 0 2 4 6 8

Time (sec)ime (sec)

Fig. 5.

p 1 = 0, he electric torqu e T is zero (excepting for a shorttransient after the first smoothed step in desired speed,when a coupling is noticed) and rotor flux dynamics reacha steady state [Fig. 2(b)]: this implies zero steady-stateerro r according to th e above analysis. This is confirmed bysimulation; we see (Fig. 2, 0 5 t 5 2) that speed and fluxsteady-state error is zero even if rotor resistance is iner ror of +50%. Starting at load insertion (at t = 2 s.), th eelectric torque and p , are different that zero which cause,according to (901, coupling and steady-state errors, asconfirmed in Figs. 2(a) and 2(b). Notice that even ifthe load torque were known (and therefore p1 = O),rotor resistance error ( p 2 # 0) would still cause a speedsteady-state error due to the entry L f 2 L f 4 1which isproportional to the electric torque [see Figs. 3(a) and3(b)]. The dynamic responses when both parameters areknown are reported in Fig. 4 for comparison.The adaptive case simulations are reported in Figs. 5. Speed and flux amplitude behavior is shown in Figs. 5(a)and 5(b), respectively, where solid lines represent actualvariables, dashed lines the corresponding reference valuesand dotted line the electric torque. In Figs. 5(c) and 5(d)load torque and rotor resistance are respectively given,where solid lines represent true parameter values anddashed lines the corresponding estimates. The Q matrixin (70) has been chosen equal to the identity matrix,the gain matrices K , an d K , have been chosen as( k u 1 , a 2 )= (900,60), ( k , , ,k b 2 )= (900,60) and the para-meter u pdate gain matrix r-' has been chosen as r- ' =diag ( l / y l , 1/y2) = diag (0 .4 ,8 X The dynamicperformances of the adaptive control law are satisfactory:no steady-state errors occur and transient responses aredecoupled, excepting for an initial short time interval.During the first speed transient, due to a wrong initialresistance estimate, a small flux error occurs. At the sametime, due to the torque required to increase speed, rotor


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MARINO et al.: ADAPTIVE INPUT-OUTPUT LINEARIZING CONTROL400 Applied voltage ua I 400 Applied voltage ua I

-400-0.5 1 1.5 2Time (sec)(a)

400 Applied voltage ua 1

.,,.. . . .. ... .... .. . .. ............,... ., . . ..-4wi 2: s 3 3:5 !Time (sec)(b)

400 Applied voltaae ua I

"""."".".",-..",- 4 4 4'5 ; 5' 5 6

Time (sec)(0

, ,StatorC,went a,503- 0-50

0 0.5 1 1.5 2Time (sec)(a)

Stator Current ia

4 4.5 5 5.5 6Time (sec)(C)

Fig. 6.

Fig. 7.

.""6 6.5 7 1.5 8Time (sec)(4

Stator Current iaI I

2 2.5 3 3. 5 4Time (sec)

(b)Stator Current ia

6 6.5 7 7.5 8Time (sec)(4

resistance estimate quickly converges to the true valueand complete decoupling is achieved. Fig. 6 shows thecontrol input signal U,. Contro l action consists in varyingamplitude and frequency of the applied voltage. Voltagesupply signals are well within the capabilities of actualinverters and therefore can be easily implemented bycurrent power electronic technology (see [91). Fig. 7 showsthe i, current waveform.As already rem arked the availability of flux measure-ments is an unrealistic assumption. However in the litera-ture several asymptotic flux observers have been p roposed[7], [8], [2], [461, which are rather sensitive to rotor resis-tance variations. In [361 a simulation study is reported fora control scheme in which the observer proposed in [46]

OF INDUCTION MOTORS 219

provides, on the basis of rotor speed and stator currentmeasurements, rotor flux and stator current estimates tothe adaptive control (591, (741, while rotor resistanceestimates are provided to the observer by the identifica-tion algorithm (74): those preliminary simulations showthat a good performance is still maintained.

VI. CONCLUSIONIn this paper we propose, for a detailed nonlinearmodel of an induction motor, an adaptive input-outputdecoupling control which has some advantages over theclassical scheme of field oriented control. With a compa-rable complexity exact decoupling between speed and fluxregulation is achieved and two critical parameters (rotorresistance and torque load) are identified by a convergingsecond-order identification algorithm. The m ain drawbackof the proposed control is the requirement of flux mea-surements. H owever, nonlinear flux observers from statorcurrents and rotor speed measurements have beenobtain ed in [46], and preliminary simulations show a satis-factory perform ance of the proposed algorithm even when

flux signals are provided by the observers given in [46].Additional research should analyze the influence ofsampling rate, truncation errors, measurement noise, sim-plifying modeling assumptions, unmodeled dynamics andsaturations. Moreover, the induction motor control prob-lem should motivate additional research on nonlinearmultivariable output feedback adaptive control, since onlysingle output systems have been so far considered in thenonlinea r adaptive litera ture (see [19] and [351).APPENDIX

Induction Motor Datastator resistance (0.18 Q )rotor resistance (0.15 0)stator currentstator flux linkagerotor currentrotor flux linkagevoltage inputangular speed (220 rad/s ) ratednum ber of pole pairs 1angle of rotationstato r inductanc e (0.0699 H )rotor inductance (0.0699 H )mutua l inductan ce (0.068 H )rotor inerti a (0.0586 Kgm2)load torque (70 N m ) ratedelectric motor to rque

(1.3 Wb) rated

(rated power 15 Kw)

The changes of variables which transform the motorequation in the original three phase system to the equiva-lent two phase reference frame (see [23, p. 135 and 1701)



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IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 38, NO. 2, FEBRUARY 1993

2 Tcos p - -2?r-sin p --3

12 --

220

are given by:

[21] D. Kim, I. Ha., and M. KO, Control of induction motors viafeedback linearization with input-output decoupling, Intemat. J.Contr., vol. 51, no. 4, pp. 863-883, 1990.1221 D. S. Kirschen, D. W. Novotny, and T.A. Lipo, Optimal efficiencycontrol of an induction motor drive, IEEE Tran s. EnergV Co nu .,vol. EC-2, no. 1, pp. 70-75, Mar. 1987.[23] P. C. Krause, Analysis of Electric Machinely. New YorkMcGraw-Hill, 1986.[24] P. C. Krause and C. H. Thomas, Simulation of symmetrical

. (91)

2r rcos p + -os p1 32r r3K , = I -sin p -sin p + -

where D is given bv (23).REFERENCES

A. Bellini, G. Figalli, and G. Ulivi, A microcomputer basedoptimal control system to reduce the effects of the parametervariations and speed measurements errors in induction motordrives, IEEE Trans. Indust. Appl., vol. IA-2, no. 1,pp. 42-50, 1986.-, Analysis and design of a microcomputer-based observer foran induction machine, Automatica, vol. 24, no. 4, pp. 549-555,1988.F. Blaschke, Das prinzip der feldorientierung, die grundlagef i r die transvector regelung von asynchronmaschienen, Siemens-Zeitschriji, vol. 45, pp. 757-760, 1971.-, The principle of field orientation applied to the newtransvector closed-loop control system for rotating field machines,Siemens-Rev, vol. 39, pp. 217-220, 1972.M. Bodson and J. Chiasson, Application of nonlinear controlmethods to the positioning of a permanent magnet stepper motor,in Proc. 28th Int. Conf Decision Contr., Tampa, FL, 1989, pp.C. C. Chan, W. S. Leung, and C. W . Ng, Adaptive decouplingcontrol of induction motor drives, IEEE Trans. Indust. Electronics,vol. 37, no. 1, pp. 41-47, Feb. 1990.Y. Dote, Existence of limit cycle and stabilization of inductionmotor via new nonlinear state observer, IEEE Trans. Automat.Contr., vol. AC-24, no. 3, pp. 421-428, June 1979.-, Stabilization of controlled current induction motor drivesystem via new nonlinear state observers, IEEE Trans. Ind. Elect.Contr., Instrum., vol. IECI-27, pp. 77-81, May 1980.-, Servo Motor and Motion Control Using Digital Signal Proces-sors.A. E. Fitzgerald, C. Kingsley, Jr., and S . D. Umans, ElectricMachinery. New York: McGraw-Hill, 1983.E. Freund, The structure of decoupled nonlinear systems, Int. J.Contr., vol. 21, no. 3, pp. 443-450, 1975.L. J. Garces, Paramete r adaptation for speed-controlled static ACdrive with a squirrel-cage induction motors, IEEE Trans. Indust.Appl., vol. IA-16, no. 12, pp. 173-178, Mar. 1980.L. R. Hunt, R. Su , and G. Meyer, Design for multiinput nonlinearsystems, in R. W . Brockett, R. S . Millman, and H. J. Sussmann,Eds., Differential Geometric Control Theory. pp. 268-298,Birkhauser, Boston, 1983.M. Ilic-Spong, R. Marino, S. Peresada, and D. G. Taylor, Feed-back linearizing control of switched reluctance motors, IEEETrans. Automat. Contr., vol. AC-32, no. 5, pp. 371-379, May 1987.A. Isidori, Nonlinear Control Systems. Communications and Con-trol Engineering Series. Berlin: Springer-Verlag, second ed., 1989.A. Isidori, A. J. Krener, C. Gori Giorgi, and S . Monaco, Nonlin-ear decoupling via feedback A differential geometric approach,IEEE Trans. Automat. Contr., vol. AC-26,pp. 331-345, 1981.

531-532.

Englewood Cliffs, NJ: Prentice-Hall, 1990.

at the 10th IFAC World Congress, pp. 349-354, Munich, 1987.A. Kusko and D. Galler, Control means for minimization oflosses in AC and DC motor drives, IEEE Trans. Indust. Appl ., vol.IA-19, no. 4, pp. 561-570, July/Aug. 1983.W. Leonhard, Control of Electrical D rives. Berlin: Springer-Verlag,1985.C. M. Liaw, C. T. Pan, and Y. C. Chen, An adaptive controll er forcurrent-fed induction motor, IEEE Trans. Aerosp. Elect. Sys t., vol.24, no. 3, pp. 250-262, 1988.A. De Luca and G. Ulivi, Dynamic decoupling of voltage fre-quency controlled induction motors, presented at the 8th Int.Conf Analysis Optimiz. Syst., pp. 127-137, INRIA, Antibes, 1988.-, Design of exact nonlinear controller for induction motors,IEEE Trans. Automat. Contr., vol. 34, no. 12, pp. 1304-1307, Dec.1989.R. Marino, An example of nonlinear regulator, IEEE Trans.Automat. Contr.,vol. AC-29, pp. 276-279, Mar. 1984.-, On the largest feedback linearizable subsystem, Syst. &[email protected]. , vol. 6, pp. 345-351, Jan. 1986.R. Marino, S. Peresada, and P. Valigi, Adaptive partial feedbacklinearization of induction motors, in Roc . 29 th Conf DecisionContr., pp. 3313-3318, Honolulu, HI, 1990.-, Adaptive nonlinear control of induction motors via extendedmatching, P. V. Kokotovic, Ed., Foundations of Adaptitre Control,(Lecture Notes in Control and Inf. Sciences). Berlin: Springer-Verlag, pp. 435-454, 1991.R. Marino and P. Tomei, Global adaptive observers and output-feedback stabilization for a class of nonlinear systems, P. V.Kokotovic, Ed., Foundations of Adaptiue Control, (Lecture Notes inControl and Inf. Sciences). Berlin: Springer-Verlag, pp. 455-493,1991.R. Marino and P. Valigi, Nonlinear control of induction motors: Asimulation study, in Proc. 1991 European Contr. Conf ,Grenoble,France, 1991, pp. 1057-1062.G. Meyer, R. Su , and L. R. Hunt, Application of nonlineartransformation to automatic flight control, Automatica, vol. 20,no. 1, pp. 103-107, 1984.K. Nam and A. Aropostathis, A model reference adaptive cont rolscheme for pure feedback nonlinear systems, IEEE Trans.Automat. Contr.,vol. 33, pp. 803-811, 1988.H. Nijmeijer and A. J. van der Schaft, Nonlinear Dynamical ControlSystems. Berlin: Springer-Verlag, 1990.V. H. Popov, Hyperstability of Control Systems. Berlin: Springer-Verlag, 1973.L. Praly, G. Bastin, J. B. Pomet, and Z. P. Jiang, Adaptivestabilization of nonlinear systems, P. V. Kokotovic, Ed., Founda-tions of Adaptbe Control (Lecture Noes in Control and Inf. Sci-ences). Berlin: Springer-Verlag, pp. 347-433, 1991.


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MARINO et al.: ADAPTIVE INPUT-OUTPUT LINEARIZING CONTROL OF INDUCTION MOTORS 221[42] A. Sabanovic and D. Izosimov, Application of sliding modes toinduction motor control, IEEE Trans.Indust.Appl. , vol. IA-17, no.1, pp. 41-49, Jan./Feb. 1981.[43] S. S. Sastry and A. Isidori, Adaptive control of linearizablesystems, IEEE Trans. Automat. Contr., vol. 34, pp. 1123-1131,1989.[44] D. G. Taylor, P. V. Kokotovic, R. Marino, and I. Kanellakopoulos,Adaptive regulation of nonlinear systems with unmodeled dynam-ics, IEEE Trans. Automat. Contr.,vol. 34, pp. 405-412, 1989.[45] A. Teel, R. Kadiyala, P. Kokotovic, and S. S. Sastry, Indirect

techniques for adaptive input-output linearization of nonlinearsystems, Int. J. Contr.,vol. 53, pp. 193-222, 1991.[46] G. C. Verghese and S. R. Sanders, Observers for flw estimationin induction machines, IEEE Trans. Indust. Elect., vol. 35, no. 1,pp. 85-94, Feb. 1988.

Riccardo Marino was born in Ferrara, Italy, in1956. He received the degree in nuclear engi-neering, in 1979, and the Masters degree insystems engineering, in 1981, both from theUniversity of Rome La Sapienza, Rome, Italy.He received the Doctor of Science degreein system science and mathematics fromWashington University, St. Louis, MO, in 1982.Since 1984, he has been with the Departmentof Electronic Engineering at the University ofRome Tor Vergata, Rome, Italy where he iscurrently Professor of systems theory. He visited the University ofIllinois at Urbana-Champaign, IL, during the academic years of 1985-86and 1988-89 and the University of Twente, The Netherlands, in 1986.His research interests include theory and applications of nonlinearcontrol and, more recently, nonlinear adaptive control.

Sergei M. Peresada was bom in Donetsk, USSR,on January 14, 1952. He received the Diplomaof Electrical Engineer from Donetsk Polytechni-cal Institute, Donetsk, in 1974 and the Candi-date of Sciences degree in electrical engineeringfrom the Kiev Polytechnical Institute, Kiev, in1983.From 1974 to 1977 he was a Research Engi-neer in the Department of Electrical Engineer-ing, Donetsk Polytechnical Institute. Since 1977he has been with the Department of ElectricalEngineering, Kiev Polytechnical Institute, Ukraine, where he is currentlyDocent (the equivalent of Associate Professor in the US). From 1985 to1986 he was a Visiting Professor in the Department of Electrical andComputer Engineering, University of Illinois at Urbana, Champaign. Hisresearch interests are applications of modem control theory (nonlinearcontrol, adaptation, VSS control) in electromechanical systems, modeldevelopment, and control of electrical drives and internal combustionengines.

Paolo Valigi received the B.S. degree in elec-tronic engineering from the University of RomeLa Sapienza, in 1986 and the Ph.D. degreefrom the University of Rome Tor Vergata, in1991.From 1986 to 1991 he has been at the Depart-ment of Electronic Engineering, University ofRome Tor Vergata. In 1989 he was a visitingscholar, at the Coordinated Science Laboratory,University of Illinois at Urbana Champaign. Hisresearch interests are in nonlinear control,robust, and adaptive control, queueing systems, and robotics.

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1993 Marino TAC IM Feedback Linearization